Question

If 4 is zero of the cubic polynomial x³ – 3x² – 10x + 24, find its other two zeroes.

Answer 1

x³ – 3x² – 10x + 24

Let α, β and γ be the zeroes of given polynomial

∴ α + β + γ = 3   ...(i)

αβ + βγ + γα = –10   ...(ii)

αβγ = –24   ...(iii)

Given: α = 4 from equation (i) β + γ = – 1

From equation (ii) βγ = – 6

(β – γ)² = (β + γ)² – 4βγ

= (–1)² – 4(–6)

= 25

∴ β – γ = ±5

β – γ = 5   ...(iv)

β + γ = –1

2β = 4

⇒ β = 2

And γ = – 3

Hence, zeroes are –3, 2 and 4.